More advanced aspects of the Dyson equation

 

The Dyson equation is one of the central relations in quantum many-body theory. It connects a reference Green’s function, which describes propagation without the many-body corrections of interest, to the full interacting Green’s function.

Its main physical role is to describe how interactions modify the propagation of a particle. Instead of moving as an ideal noninteracting particle, an electron interacts with its environment, polarizes the surrounding medium, scatters from other excitations, and acquires a modified energy and a finite lifetime.

All these effects are collected into the self-energy operator.

Reference and interacting Green’s functions

Consider a reference one-particle Hamiltonian

$$\widehat{H}_0.$$

The corresponding reference Green’s function is

$$
G_0(z) = \left[z\widehat{I}-\widehat{H}_0\right]^{-1},$$

where $z$ is a complex energy or frequency variable.

For a retarded Green’s function, one usually writes

$$z=\omega+i\eta, \qquad \eta\rightarrow 0^+.$$

Therefore,

$$G_0^{R}(\omega) = \left[ (\omega+i\eta)\widehat{I}-\widehat{H}_0 \right]^{-1}.$$

The infinitesimal positive term $i\eta$ specifies the retarded boundary condition and determines the analytic structure of the Green’s function.

The reference Hamiltonian does not necessarily describe a completely free electron. Depending on the application, $\widehat{H}_0$ may represent:

  • a truly noninteracting particle;
  • an electron moving in an external potential;
  • a Hartree or Hartree–Fock reference system;
  • a Kohn–Sham Hamiltonian obtained from density functional theory.

Interactions beyond the chosen reference system modify the propagator. Their effect is represented by the self-energy operator

$$\Sigma(z).$$

The full Green’s function is then defined by

$$G(z)=\left[z\widehat{I}-\widehat{H}_0-\Sigma(z) \right]^{-1}.$$

Unlike an ordinary static potential, the self-energy may be nonlocal, frequency dependent, complex, and non-Hermitian.

Inverse form of the Dyson equation

The inverse reference Green’s function is

$$G_0^{-1}(z)=z\widehat{I}-\widehat{H}_0.$$

The inverse interacting Green’s function is

$$G^{-1}(z) = z\widehat{I}-\widehat{H}_0-\Sigma(z).$$

Therefore,

$$G^{-1}(z) = G_0^{-1}(z)-\Sigma(z)$$

This is the inverse form of the Dyson equation.

It shows that the self-energy modifies the inverse propagator rather than simply adding an independent propagation channel.

Matrix form

In a discrete basis of orbitals, bands, atomic states, or other basis functions, the Dyson equation becomes

$$G_{ab}(\omega) = G_{0,ab}(\omega) + \sum_{cd} G_{0,ac}(\omega) \Sigma_{cd}(\omega) G_{db}(\omega).$$

In matrix notation,

$$\mathbf G(\omega) = \mathbf G_0(\omega) + \mathbf G_0(\omega) \boldsymbol\Sigma(\omega) \mathbf G(\omega)$$

The equivalent inverse form is

$$\mathbf G(\omega) = \left[ 
\mathbf G_0^{-1}(\omega) - \boldsymbol\Sigma(\omega) \right]^{-1} $$

If the reference Green’s function is written as

$$ \mathbf G_0(\omega) = \left[ (\omega+i\eta)\mathbf I-\mathbf H_0 \right]^{-1}, $$

then

$$\mathbf G(\omega) = \left[(\omega+i\eta)\mathbf I - \mathbf H_0- \boldsymbol\Sigma(\omega) \right]^{-1}.$$

This is the form commonly used in practical many-body calculations.

Operator form of the Dyson equation

The compact operator form of the Dyson equation is

$$G^{-1} = G_0^{-1}-\Sigma.$$

Equivalently,

$$G = \left( 
G_0^{-1}- \Sigma \right)^{-1}.$$

This form is especially useful in matrix calculations and numerical implementations. In a chosen basis,

$$\mathbf G(\omega) = \left[ 
\mathbf G_0^{-1}(\omega) \boldsymbol\Sigma(\omega) \right]^{-1}.$$

Because these quantities are operators or matrices, the order of multiplication matters. Products such as $G_0\Sigma G$ represent operator products or convolutions over intermediate variables.

Alternative derivation using the resolvent identity

The Dyson equation can also be derived directly from the resolvent form of the Green’s function.

The reference Green’s function is

$$G_0 = \left( E-H_0 \right)^{-1},$$

while the full Green’s function is

$$G = \left( E-H_0-\Sigma \right)^{-1}.$$

We begin by inserting the identity operator in the form

$$G_0 \left( 
E-H_0 \right) = I.$$

Multiplying the full Green’s function by this identity gives

$$G = G_0 \left( E-H_0 \right) G.$$

Now add $-\Sigma+\Sigma$ in the parenthesis and separating the two terms gives

$$G = G_0 \left( E-H_0-\Sigma \right) G + G_0\Sigma G.$$

Since

$$\left( 
E-H_0-\Sigma \right) G = I, $$

the first term becomes $G_0$. Therefore,

$$G = G_0 + G_0\Sigma G $$

This is the Dyson equation.

This derivation is an example of the operator resolvent identity and is useful when working directly with Hamiltonians and inverse operators.

When does a dressed particle become a quasiparticle?

A dressed excitation can be described as a quasiparticle only if it remains sufficiently long-lived and produces a well-defined peak in the spectral function.

This requires the imaginary part of the self-energy to be sufficiently small near the excitation energy. The quasiparticle is then characterized by its renormalized energy, finite lifetime, and quasiparticle weight $Z$.

If the spectral peak becomes too broad or $Z$ becomes very small, the quasiparticle picture may break down.

Energy shifts and finite lifetimes

For a translationally invariant system with a single relevant band, the Green’s function can be written schematically as

$$G(\mathbf k,\omega) = \frac{1}{\omega - \varepsilon_{\mathbf k}^{0} - \Sigma(\mathbf k,\omega)}.$$

The self-energy is generally complex:

$$\Sigma(\mathbf k,\omega) = \operatorname{Re}\Sigma(\mathbf k,\omega) + i\operatorname{Im}\Sigma(\mathbf k,\omega).$$

Its real and imaginary parts have different physical meanings.

The real part,

$$\operatorname{Re}\Sigma(\mathbf k,\omega), $$

shifts and renormalizes the excitation energy.

The imaginary part,

$$\operatorname{Im}\Sigma(\mathbf k,\omega),$$

produces spectral broadening and is related to the finite lifetime of the excitation.

A stable noninteracting state would produce an infinitely sharp pole. In an interacting system, scattering and decay processes broaden this pole into a spectral peak with a finite width.

The quasiparticle energy approximately satisfies

$$E_{\mathbf k} = \varepsilon_{\mathbf k}^{0} + \operatorname{Re} \Sigma(\mathbf k,E_{\mathbf k}).$$

More generally, if the reference Hamiltonian already contains an exchange-correlation potential or another effective contribution, the corresponding double-counting term must be treated consistently.

The quasiparticle lifetime is related to the imaginary part of the self-energy. Schematically,

$$\frac{1}{\tau_{\mathbf k}} \propto \left| \operatorname{Im} \Sigma(\mathbf k,E_{\mathbf k}) \right|.$$

Thus, the Dyson equation provides both:

  • corrected excitation energies;
  • information about decay, scattering, and spectral linewidths.

What the Dyson equation does not determine

The Dyson equation itself does not specify the self-energy.

It only states how a given self-energy modifies the Green’s function:

$$G^{-1} = G_0^{-1} \Sigma.$$

To calculate $G$, one must choose an approximation for $\Sigma$.

Different approximations retain different classes of many-body processes. The accuracy of the resulting Green’s function therefore depends strongly on the approximation used for the self-energy.

Examples include:

  • Hartree approximation;
  • Hartree–Fock approximation;
  • second-order perturbative self-energies;
  • electron-phonon self-energies;
  • dynamical mean-field theory self-energies;
  • the GW approximation.

Connection to the GW approximation

In the GW approximation, the self-energy is constructed from the Green’s function $G$ and the screened Coulomb interaction $W$.

Schematically,

$$\Sigma \approx iGW $$

The Green’s function describes particle propagation, while $W$ describes the dynamically screened interaction produced by the response of the electronic system.

The Dyson equation then becomes part of a connected set of many-body equations:

$$G \longrightarrow \Sigma \longrightarrow G.$$

In a self-consistent calculation, the Green’s function is used to construct the self-energy, and the resulting self-energy is inserted back into the Dyson equation to obtain an updated Green’s function.

In the widely used one-shot $G_0W_0$ approximation, the self-energy is instead evaluated from a fixed reference Green’s function $G_0$ and a corresponding screened interaction $W_0$.

Before discussing the GW approximation in detail, the next section examines the physical and diagrammatic meaning of the self-energy itself.